Solve for $x$ : $ 2|x - 7| + 9 = 1|x - 7| + 10 $
Solution: Subtract $ {1|x - 7|} $ from both sides: $ \begin{eqnarray} 2|x - 7| + 9 &=& 1|x - 7| + 10 \\ \\ { - 1|x - 7|} && { - 1|x - 7|} \\ \\ 1|x - 7| + 9 &=& 10 \end{eqnarray} $ Subtract ${9}$ from both sides: $ \begin{eqnarray} 1|x - 7| + 9 &=& 10 \\ \\ { - 9} &=& { - 9} \\ \\ 1|x - 7| &=& 1 \end{eqnarray} $ Simplify: $ |x - 7| = 1$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 7 = -1 $ or $ x - 7 = 1 $ Solve for the solution where $x - 7$ is negative: $ x - 7 = -1 $ Add ${7}$ to both sides: $ \begin{eqnarray} x - 7 &=& -1 \\ \\ {+ 7} && {+ 7} \\ \\ x &=& -1 + 7 \end{eqnarray} $ $ x = 6 $ Then calculate the solution where $x - 7$ is positive: $ x - 7 = 1 $ Add ${7}$ to both sides: $ \begin{eqnarray} x - 7 &=& 1 \\ \\ {+ 7} && {+ 7} \\ \\ x &=& 1 + 7 \end{eqnarray} $ $ x = 8 $ Thus, the correct answer is $x = 6 $ or $x = 8 $.